19

Chapter 3

Design for manufacture of Fabry-Perot cavity sensors

When Fabry-Perot cavity sensors are manufactured, the thickness of each

layer must be tightly controlled to achieve the target performance of a sensor.

However, there are unavoidable errors in thickness even though techniques of

thickness control for thin films have rapidly improved [1] . For Fabry-Perot

optical interference filters it has long been recognized that the performance of the

filter is greatly influenced by random thickness variations in the films. For

instance, the resonant wavelength is very sensitive to thickness variations.

Degradation in the performance of the filters has been simulated by Monte Carlo

technique or analytic methods [2, 3] .

In this chapter the impact of manufacturing-induced variations on the

performance of Fabry-Perot cavity sensors is studied. In particular, we consider

how random variations in thickness of the cavity mirrors influence the accuracy

with which gap can be measured. Through the study, it is found that there exists

an optimum design of a Fabry-Perot cavity sensor, which is a combination of

initial cavity gap and mechanical travel of the moving mirror. The optimum

design gives large manufacturing tolerance to the thickness variations of layers

and it leads to high accuracy.

20

3.1 IMPACT OF THICKNESS VARIATIONS ON OPTICAL RESPONSE

In this section, the variation of the optical response, e.g., of reflectance,

due to the thickness variation of the layers is calculated using an analytic method.

This calculation will be a basis for designing Fabry-Perot sensors for

manufacture, which is to be discussed in the following section. The first

derivative of the reflectance ( R ) with respect to the thickness of the ith layer can

be calculated by using the first derivative of the equivalent characteristic matrix

( M ). From equation (2.4)

∂R

∂zi= ∂

∂ziP

⋅ P* + P ⋅ ∂∂zi

P*

, (3.1)

where P =η in B − C

η in B + C and P* is the conjugate of P . The partial derivatives of

B and C are calculated from

∂∂zi

B

C

= ∂∂zi

M ⋅1

η f

. (3.2)

The first derivative of the equivalent characteristic matrix is obtained using

equation (2.3)

∂M

∂zi= Mq ⋅ Mq−1 ⋅ ⋅ ∂Mi

∂zi⋅ ⋅M1, (3.3)

where ∂Mi∂zi

= ko ni−sin ko ni zi( ) j

ηicos ko ni zi( )

j ηi cos ko ni zi( ) −sin ko ni zi( )

.

21

Using equations (3.1), (3.2) and (3.3), the first derivatives of the reflectance of a

Fabry-Perot cavity with respect to each layer can be calculated.

For the Fabry-Perot cavity sensors described in chapter 2, the first

derivatives of the reflectances with respect to each layer are calculated as a

function of gap. Figure 3.1 shows a plot of the first derivative of the reflectance

of the Fabry-Perot cavity with metal (Au) mirrors. The mirrors of the cavity are

numbered in ascending order from the fiber, for convenience. From the figure, it

is obvious that the reflectance of the cavity is the most sensitive to the thickness

variation of the first Au layer which the incident light meets. Thus, thickness

control of the first Au layer would be very critical to achieve a target reflectance

of the cavity.

As shown in equation (3.3), the first derivative of reflectance with respect

to a layer also depends on the refractive index of the layer. It implies that for a

given thickness variation the reflectance curve would be more perturbed by

thickness variation of a layer with larger refractive index. Figure 3.2 plots the

first derivatives of the reflectance of a Fabry-Perot cavity with dielectric mirrors

as described in chapter 2. Each dielectric layer is numbered in ascending order

from the fiber. For example, nitride 2 represents the second silicon nitride layer

which the incident light meets. Comparing Figure 3.1 with Figure 3.2, it should

be noted that the magnitudes of the derivatives with respect to the dielectric

layers, i.e., silicon dioxide and silicon nitride, are much smaller than ones with

respect to the metal layers (Au). This indicates that a film with small refractive

index would be preferred as a mirror layer if manufacturing process for the film

22

produces the same process-induced thickness variation as a film with large

refractive index, assuming the layers were of equal thickness.

Figure 3.1 : The first derivatives of the reflectance of a Fabry-Perot cavity withmetal (Au) mirrors, as shown in Figure 2.5. The vertical axisrepresents the first derivative of the reflectance of the cavity withrespect to the thickness of each layer.

23

Figure 3.2 : The first derivatives of the reflectance of a Fabry-Perot cavity withdielectric films, as shown in Figure 2.6. For simplicity, only thefour layers with the largest derivatives are shown.

24

3.2 DESIGN METHODOLOGY AND CASE STUDY FOR MANUFACTURE

If Fabry-Perot sensors are to be manufactured in large volume at low cost,

it will not be practical to individually calibrate every sensor to eliminate

fabrication-induced response variations. Instead of calibrating every sensor, it

would be preferable if a nominal response curve could be given for a set of the

manufactured sensors with some specified accuracy. The accuracy is limited by

the amount of change in the response curve due to the thickness variation of the

layers.

Quantitative treatment of the accuracy of a design is discussed in the

following subsections. Using the first derivatives obtained in the last section,

uncertainty in gap induced by thickness variation can be calculated and included

in the calculation of accuracy. Finally, the accuracy can be used as a metric to

find an optimum design for manufacture of Fabry-Perot cavity sensors.

3.2.1 Design Methodology

For Fabry-Perot sensors, the cavity gap g could be inferred from some

measurement of reflected light intensity I, that is a function of the reflectance R,

that is in turn a function of wavelength and the structure of the sensor. For

example, absolute reflectance of the cavity could be used as a measurand as done

in chapter 2. The detection scheme is, however, very susceptible to any light

intensity change in the fiber due to coupling and fiber bending. As an alternative,

a ratio of the reflectances at two wavelengths (λ1 and λ2) could be used as a

measurand. Such a detection scheme will be called the dual wavelength

25

technique in this chapter. This method eliminates errors resulting from

wavelength-independent changes in the fiber interconnect to the sensor, such as

fiber bending loss and coupling loss.

Taking manufacturing-induced variations of layer thickness into account,

it should be realized that a given measurand value I R,λ( ) could be produced

from a Fabry-Perot cavity which has different layer thicknesses from a target

design. This leads to an error in estimating the cavity gap g. In other words, there

exists an uncertainty ∆g in the gap at a given measurand I R,λ( ) since for mirror

layers with thicknesses z1 + ∆z1, z2 + ∆z2 ,⋅ ⋅⋅, zq + ∆zq( ) there can exist a gap

thickness g + ∆g that would produce an identical value of I

I R z1, z2 ,⋅ ⋅⋅,g,⋅ ⋅⋅, zq , λ( )[ ] = I R z1 + ∆z1, z2 + ∆z2 ,⋅ ⋅⋅,g + ∆g,⋅ ⋅⋅, zq + ∆zq , λ( )[ ] (3.4)

where the functional dependencies of R have been indicated explicitly.

In principle, it should be possible to obtain ∆g by calculating the response

for all possible thickness combinations weighted by the distribution functions

representing the process-induced thickness variation of each layer. This approach

is computationally impractical when q becomes large (for instance, if the cavity

mirrors are made using multilayer dielectric mirrors).

The uncertainty ∆g in gap can be analytically expressed using a Taylor

series approximation

26

I R z1 + ∆z1, z2 + ∆z2 ,⋅ ⋅⋅,g + ∆g,⋅ ⋅⋅, zq + ∆zq , λ( )[ ]≈ I R z1, z2 ,⋅ ⋅⋅,g,⋅ ⋅⋅, zq , λ( )[ ] + ∂ I

∂ zi

⋅ ∆zi +i =1i ≠ k

q∑ ∂ I

∂ g

⋅ ∆g

.(3.5)

Combining equations (3.4) and (3.5) then gives ∆g;

∆g ≈ − ∂ I∂ g

−1

⋅ ∆zi ⋅ ∂ I∂ zi

i =1

i ≠ k

q

∑ . (3.6)

To bound the uncertainty in gap, we should find the combination of layer

thicknesses that maximizes ∆g for a given set of ∆zi; examination of equation

(3.6) gives this bound, ∆gproc, as

∆gproc = ∂ I∂ g

−1 ∂ I∂ zi

⋅ ∆zii =1i ≠ k

q

∑ . (3.7)

In terms of yield, if for each mirror layer i (i    k) the fraction of devices

with thickness between zi − ∆zi and zi + ∆zi is Pi, then for a fixed value of

I R,λ( ), the fraction of sensors P∆g with gap between g − ∆gproc and g + ∆gproc

that would produce this value of I R,λ( ) would be at least

P∆g = Pii=1i≠k

q∏ . (3.8)

To further specify the performance of the Fabry-Perot sensor

quantitatively, the operational range for the device must be specified. Here we

assume two primary design space variables: i) the initial gap gi of the Fabry-Perot

27

cavity; and ii) the maximum mechanical travel t of the moving mirror. The

second variable, t , is determined by the maximum stimulus (e.g., the maximum

pressure) and the mechanical compliance of the membrane supporting the moving

mirror. Since mechanical compliance frequently can be adjusted independently of

the thicknesses of the mirror layers by changing the lateral size, t is considered a

freely adjustable design variable. Over the full range of stimuli, g varies between

gi and gi - t , assuming the stimulus generates a motion of the mirror which

decreases the cavity gap. Also, ∆gproc varies since ∆gproc is a function of g. We

are now ready to see how uncertainty ∆gproc in the gap influences the accuracy of

the sensor: since the maximum stimulus corresponds to the maximum travel t , the

percentage accuracy would be bounded by the maximum value of ∆gproc in the

interval between gi and gi - t , divided by t .

For example, if one wishes to find the accuracy [4] for a given nominal

initial gap gi and travel t , including the uncertainties induced by thickness

variations in mirror layers, one must find the error ε proco

ε proco = max

∆ gproc

t: g ∈ gi − t , gi[ ]

. (3.9)

The best design (i.e., the best values of gi and t ) is the one that minimizes ε proco .

However, it should be remembered that manufacturing uncertainty in layer

thickness also includes the process that determines the initial gap gi. Even if one

assumes a single calibration measurement is made to determine the specific value

of gi for a given manufactured sensor, the overall design should still allow gi to

vary over the range gi.(1 ± δ ), where δ is the normalized thickness variation for

the gap layer. Thus, the accuracy defined by equation (3.9) must be modified to

28

ε proc = max∆ gproc

t: g ∈ gi ⋅ 1 − δ( ) − t , gi ⋅ 1 + δ( )[ ]

. (3.10)

The best design, taking uncertainties in the thickness of all layers into account, is

the one that minimizes ε proc .

In terms of yield, if the fraction of sensors with actual gap in the range

gi.(1 ± δ ) is Pk, then the fraction of sensors P with accuracy not worse than

ε proc is at least

P = Pk ⋅ P∆g . (3.11)

In addition to the process-induced errors, one could include errors from

fitting the actual measurand I R,λ( ) with a fitting function gfit . Since it is

usually not possible to analytically invert the function I(R) to find g, a common

approach is to find a fitting function gfit(I) to relate the gap g to the actual

measurand I. Assuming all the mirror layers take on their nominal thicknesses,

using equations (2.2)-(2.4) we can find I(g) exactly. The difference between the

exact relationship and the fitting function is then ∆ g fit

∆ gfit = g fit I g( )[ ] − g . (3.12)

If one wishes to find a fitting function that gives the best accuracy for a given

initial gap gi and travel t , we must choose a fitting function so that the error ε fit

ε fit = max∆ g fit

t: g ∈ gi − t , gi[ ]

(3.13)

is minimized.

Note that the fitting function gfit minimizing equation (3.13) is different

from the least squares fitting function gleast , which minimizes εleast

29

ε least = gleast[I(g)] − g2:g ∈ gi − t,gi[ ]

g∑

. (3.14)

Comparing the definitions of the two fitting functions, gfit would give betteraccuracy than gleast .

If both fitting and process-induced errors are included, the design that

would give the best accuracy is one where the total error εtot

εtot = max∆ g fit + ∆ gproc

t: g ∈ gi ⋅ 1 − δ( ) − t , gi ⋅ 1 + δ( )[ ]

(3.15)

is minimized.

30

3.2.2 Example Designs

To illustrate a design process, the measurand I R,λ( ) must be specified. If

reflectance of the cavity at a single wavelength λ1 is used as the measurand,

I R z1, z2 ,⋅ ⋅⋅,g,⋅ ⋅⋅, zq , λ( )[ ] = R z1, z2 ,⋅ ⋅⋅,g,⋅ ⋅⋅, zq , λ1( ) . (3.16)

This response curve will be periodic in g, with period λ1/2 and equation (3.7)

becomes

∆gproc = ∂ R∂ g

−1 ∂ R∂ zi

⋅ ∆zii =1i ≠ k

q

∑ , (3.17)

where R can be found using the method discussed in chapter 2.

Figure 3.3 shows reflectance and ∆gproc of a Fabry-Perot sensor with

metal (Au) mirrors described in the chapter 2. The periodicity of the curve

suggests two basic operating branches, one between 100 nm and 275 nm, and the

other between 275 nm and 450 nm. Using equation (3.17) ∆gproc was calculated,

assuming thickness variation ∆z for the Au layers is ±3 Å (three sigma), which

can be achieved with thin film coating equipment.

For comparison, extensive random combinations of Au mirror thickness

from 67 Å to 73 Å have been tested to verify that for a given reflectance R the

maximum change in g is produced by the perturbed layer thicknesses used in

equation (3.7). Agreement between the two approaches indicates that the first

order Taylor series approximation used in equation (3.5) is sufficient for this case.

31

Figure 3.3 : Reflectance and process-induced response variations of a Fabry-Perot cavity with Au mirrors for a single wavelength (lambda = 700nm). Solid line: reflectance; Dotted line: bound on gap uncertainty∆gproc.

For dielectric mirrors, the variation in thickness of each layer was assumed

to be ±3% of the nominal thickness. State-of-the-art deposition or sputtering

technology can provide ±3% variation in thickness with ±3σ process tolerance

[5, 6]. Figure 3.4 shows a plot of reflectance and ∆gproc of the Fabry-Perot cavity

with the dielectric layers described in the chapter 2. Even though the dielectric

layers show the smaller first derivatives as shown in Figure 3.2, the maximum

uncertainty in the gap (∆gproc) is much bigger than one with metal mirrors. The

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1

1

10

100

1000

0 1000 2000 3000 4000 5000

Ref

lect

ance

Gap

Unc

erta

inty

Bou

nd (

Å)

Cavity Gap (Å)

32

bigger maximum uncertainty is caused by thickness variation, which is

proportional to the nominal thickness of the dielectric layers, as indicated in

equation (3.17).

Figure 3.4 : Reflectance and process-induced response variations of a Fabry-Perot cavity with dielectric mirrors for a single wavelength (lambda= 700 nm). Solid line: reflectance; Dotted line: bound on gapuncertainty ∆gproc.

°°°°

°

°

°

°

°°°°°°°°°°°°°°°°°°°°°°°°°°°

°

°

°

°

°°°°°°°°°°°°°°°°°°°°°

°°°°°°

°

°

°

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

100

300

500

700

900

1100

500 1500 2500 3500 4500 5500

Ref

lect

ance

Gap

Unc

erta

inty

Bou

nd (

Å)

Cavity Gap (Å)

33

Note that for both the Fabry-Perot cavities the process-induced error

∆gproc in one branch is smaller than in the other. This implies that although the

response curves for the two branches are almost identical, their sensitivity to layer

thickness variations is not.

To find an optimum design which produces the best accuracy contour

maps can be generated over the design space, i.e., for each choice of nominal

initial gap gi and maximum mechanical travel t . For simplicity, we first assume

it is desirable to find a design that produces the best accuracy when the only

errors are due to layer thickness variations (i.e., we use a “high order” fitting

function so that ∆ g fit = 0). Equation (3.10) is then used to find ε proc for each

possible design and contours of constant accuracy are plotted against gi and t .

As an illustration contour maps are generated for the case of metal mirrors

because ∆gproc of dielectric mirrors is much worse than one of metal mirrors as

shown in Figure 3.4 and Figure 3.3.

Figure 3.5 shows these contours for the case where the measurand is the

single wavelength reflectance of the Fabry-Perot cavity with metal (Au) mirrors.

For instance, 5 % accuracy could be achieved for a range of designs with gi

between about 2470 Å and 2700 Å, with corresponding travel between about 200

Å and 1200 Å. The fraction of devices with accuracy not worse than the contour

value is given by equation (3.11); since we have used three sigma values for the

layer thickness variations (Pi = 0.99), and there are three layers in this example

(two metal mirrors plus the sacrificial layer), the fraction of sensors P with

accuracy not worse than ε proc is [0.99] 3, i.e., at least 97 %.

34

Figure 3.5 : Contour map over design space (i.e. initial gap and mechanicaltravel) when only process-induced thickness variations areconsidered. The numbers on the contour lines represent theaccuracy of corresponding designs.

4100

3850

3600

3350

3100

2850

2600

2350

2100

1850

100 300 500 700 900 1100 1300 1500

Nom

inal

Initi

al G

ap (

Å)

Mechanical Travel (Å)

0.05

0.1

0.1

0.3

0.2

0.2

0.3

35

Figure 3.6 shows contours for the case where there is only fitting error. A

linear fitting function defined in equation (3.13) is assumed. Finally, Figure 3.7

shows the accuracy contour plot assuming that both process-induced variations

and linear fitting errors are included. Not surprisingly, the range of travels that

can still give 5 % accuracy is considerably reduced, with an optimum design at gi

2600 Å and t 300 Å. With this design at least 97 % of the manufactured

sensors would give 5 % accuracy or better.

Figure 3.6 : Accuracy contour map, including only linear fitting errors, for singlewavelength detection. Numbers on lines represent the accuracy ofthe contour.

4250

3750

3250

2750

2250

1750

1250

100 400 700 1000 1300 1600

Nom

inal

Initi

al G

ap (

Å)

Mechanical Travel (Å)

0.02

0.02

0.04

0.04

0.06

0.08

36

Figure 3.7 : Accuracy contour map, including both mirror variations and linearfitting errors, for single wavelength detection.

3750

3500

3250

3000

2750

2500

2250

2000

1750

100 300 500 700 900 1100

Nom

inal

Initi

al G

ap (

Å)

Mechanical Travel (Å)

0.2

0.1

0.1

0.1

0.2

0.05

37

This design process can be applied to the case where another measurand is

chosen. Suppose a ratio of two reflectances at different wavelengths λ1 and λ2 is

used as a measurand. The following equation is one of the possible ratiometric

methods

I R( ) =R λ1( )

R λ1( ) + R λ2( ) . (3.18)

Substituting I R,λ( ) in equation (3.7) with equation (3.18), ∆gprocs becomes

∆gprocs = i =1i ≠ k

q

∑ R2∂R1

∂ zi− R1

∂R2

∂ zi

⋅ ∆zi

R1∂R2

∂ g− R2

∂R1

∂ g

. (3.19)

Figure 3.8 shows the response curve and associated process-induced errors

of the cavity with metal (Au) mirrors for dual wavelength detection, assuming

detection wavelengths of 560 nm and 700 nm. The response curve is still a

periodic function with respect to gap, but with a period equal to the lowest

common multiple of λ 1

2 and

λ 2

2 . There are now nine distinct operating

branches. Note that the process-induced errors in the branch from 8000 Å to

10000 Å are smaller than any other regions.

Figure 3.9 shows contour maps including only process-induced errors.

The two best branches are chosen which have smaller process-induced errors than

any other branches as shown in Figure 3.8. The maps suggest an optimum design

for gi = 9325 Å and t = 825 Å, with at least 97 % of the sensors producing an

accuracy of not worse than 1 %.

38

Figure 3.8 : Response curve and process-induced variations of a Fabry-Perotcavity with metal mirrors for dual wavelengths (560nm and700nm).Solid line: response curve; Dashed line: process-induced responsevariations.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

100

1000

0 3000 6000 9000 12000 15000

Dua

l wav

elen

gth

sign

al

Gap

Unc

erta

inty

Bou

nd (

Å)

Cavity Gap (Å)

39

Contour maps including only linear fitting-induced error indicate that an

accuracy not worse than 1 % could be achieved using gi = 6250 Å and t = 1050

Å, as shown in Figure 3.10. Note that when using dual wavelength detection

linearity is maintained over a much longer travel than for single wavelength

detection, as has been noted in [7] .

Figure 3.11 shows the contour plots of the error induced by both linear

fitting and thickness variation for the branches with best performance. The map

near 9000 Å shows that the accuracy of this branch is heavily degraded by the

linear fitting-induced errors. When both errors are considered, the optimum

design is gi = 6050 Å and t = 650 Å, with at least 97 % of the manufactured

sensors producing an accuracy of not worse than 3.5 %.

40

Figure 3.9 : Accuracy contour maps including only process-induced errors fortwo branches with best performance.

6100

6000

5900

5800

5700

5600

5500

100 200 300 400 500 600 700

Initi

al g

ap (Å

)

Mechanical travel (Å)

0.08

0.06

0.04

0.12

0.120.08

0.06

0.04

9500

9300

9100

8900

8700

8500

100 300 500 700 900 1100

Initi

al g

ap (

Å)

Mechanical travel (Å)

0.02

0.040.06

0.1

0.02

0.04

0.06

41

6100

6000

5900

5800

5700

5600

5500

5400

100 200 300 400 500 600 700

Initi

al g

ap (

Å)

Mechanical travel (Å)

0.020.01

0.02

0.03

0.03

9500

9300

9100

8900

8700

8500

8300

100 300 500 700 900 1100

Initi

al g

ap (

Å)

Mechanical travel (Å)

0.02

0.04

0.06

0.08

0.1

Figure 3.10 : Accuracy contour maps including linear fitting errors for twobranches with best performance.

42

Figure 3.11 : Accuracy contour maps including both process-induced errors andlinear fitting errors for two branches with best performance.

6100

6000

5900

5800

5700

5600

5500

100 200 300 400 500 600 700 800

Initi

al g

ap (

Å)

Mechanical travel (Å)

0.05

0.050.1

0.1

0.2

0.2

9500

9300

9100

8900

8700

8500

100 300 500 700 900 1100

Initi

al g

ap (

Å)

Mechanical travel (Å)

0.05

0.1

0.2

0.1

43

3.3 SUMMARY

The impact of thickness variations in layers on the performance of Fabry-

Perot cavity sensors has been studied. It was found that for a given cavity there

exists an optimum design, a combination of initial gap and mechanical travel, that

gives the least variation in response curve. Through proper design high

manufacturing yield with reasonable accuracy can be achieved. For the specific

example of a Fabry-Perot pressure sensor discussed here, even with 3 % (three

sigma) layer thickness variations, better than 5 % accuracy can be achieved from

at least 97 % of the manufactured sensors by choosing the proper nominal initial

gap and associated maximum mechanical travel. Also note that since a proper

design is dependent on the choice of measurand, the measurand, i.e. detection

method, should be determined before an optimum combination of nominal initial

gap and mechanical travel is selected.

The models and design methodology discussed can also be extended easily

to other sources of manufacturing variation, such as dielectric constant

fluctuation. Finally, it should be straightforward to apply our approach to other

interference-based micromachined devices, such as tunable interferometers [8]

and modulators [9] , to find designs that minimize device sensitivity to process-

induced variations.